Triple linkage of quadratic Pfister forms

Abstract

Given a field F of characteristic 2, we prove that if every three quadratic n-fold Pfister forms have a common quadratic $$(n-1)$$ -fold Pfister factor then $$I_q^{n+1} F=0$$ . As a result, we obtain that if every three quaternion algebras over F share a common maximal subfield then u(F) is either 0, 2 or 4. We also prove that if F is a nonreal field with $${\text {char}}(F) \ne ~2$$ and $$u(F)=4$$ , then every three quaternion algebras share a common maximal subfield.